What is Thresholding?

• Definition: A technique to separate objects from the background in images.
• Concept: Converting a grayscale image into a binary image.
• Importance: Simplification for further analysis (e.g., edge detection, segmentation).

Global Thresholding

• Explanation of Global Thresholding.
• Example of a fixed threshold ( T ):
  • If ( I(x, y) > T ), the pixel becomes white (1), otherwise black (0).
• Limitations: Sensitivity to uneven lighting.

Adaptive Thresholding

• Definition of Adaptive Thresholding.
• Instead of a global threshold, the threshold is calculated for different regions of the image.
• Advantages: Effective in images with varying illumination.
• Algorithm: Example of an adaptive method based on the local mean of neighboring pixels.

Justification for Using Thresholding

• When thresholding is useful:
  • Images with a clear contrast between the object and the background.
  • Situations requiring quick segmentation.
• Example applications:
  • Text detection, object recognition, medical images (e.g., X-rays).
• Limitations: Less effective in noisy or low-quality images.

Comparison of Thresholding Algorithms

Practical Examples

• Show examples of original images and the result after applying:
  • Global Thresholding.
  • Adaptive Thresholding.
  • Otsu’s Algorithm.
• Visualizations highlighting the differences.

Practical Examples

plt.imshow(image_circle, cmap="gray")
plt.axis("off")
plt.show()

plt.imshow(image_gradient, cmap="gray")
plt.axis("off")
plt.show()

plt.imshow(noisy_circle, cmap="gray")
plt.axis("off")
plt.show()

_, global_thresh1 = cv2.threshold(image_circle, 127, 255, cv2.THRESH_BINARY)
plt.imshow(global_thresh1, cmap="gray")
plt.axis("off")
plt.show()

_, global_thresh2 = cv2.threshold(image_gradient, 127, 255, cv2.THRESH_BINARY)
plt.imshow(global_thresh2, cmap="gray")
plt.axis("off")
plt.show()

_, global_thresh3 = cv2.threshold(noisy_circle, 127, 255, cv2.THRESH_BINARY)
plt.imshow(global_thresh3, cmap="gray")
plt.axis("off")
plt.show()

adaptive_thresh1 = cv2.adaptiveThreshold(image_circle, 255, cv2.ADAPTIVE_THRESH_MEAN_C, cv2.THRESH_BINARY, 11, 2)
plt.imshow(adaptive_thresh1, cmap="gray")
plt.axis("off")
plt.show()

adaptive_thresh2 = cv2.adaptiveThreshold(image_gradient, 255, cv2.ADAPTIVE_THRESH_MEAN_C, cv2.THRESH_BINARY, 11, 2)
plt.imshow(adaptive_thresh2, cmap="gray")
plt.axis("off")
plt.show()

adaptive_thresh3 = cv2.adaptiveThreshold(noisy_circle, 255, cv2.ADAPTIVE_THRESH_MEAN_C, cv2.THRESH_BINARY, 11, 2)
plt.imshow(adaptive_thresh3, cmap="gray")
plt.axis("off")
plt.show()

_, otsu_thresh1 = cv2.threshold(image_circle, 0, 255, cv2.THRESH_BINARY + cv2.THRESH_OTSU)
plt.imshow(otsu_thresh1, cmap="gray")
plt.axis("off");
plt.show()

_, otsu_thresh2 = cv2.threshold(image_gradient, 0, 255, cv2.THRESH_BINARY + cv2.THRESH_OTSU)
plt.imshow(otsu_thresh2, cmap="gray")
plt.axis("off")
plt.show()

_, otsu_thresh3 = cv2.threshold(noisy_circle, 0, 255, cv2.THRESH_BINARY + cv2.THRESH_OTSU)
plt.imshow(otsu_thresh3, cmap="gray")
plt.axis("off")
plt.show()

Conclusion and Questions

• Summary of key points:
  • Thresholding as a simple yet powerful technique.
  • Importance of choosing the right algorithm depending on the context.
  • Otsu’s algorithm as an effective solution for bimodal images.

Introduction to Morphological Operations

• Definition: Morphological operations apply a structuring element to an image to alter its structure.
• Focus: Primarily used for binary images.
• Key applications: Noise removal, object extraction, shape analysis.

Structuring Element

• A small matrix used to probe and interact with a given image.
• Common shapes: Rectangular, circular, elliptical.
• Example: A 3x3 square structuring element.

\[\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}\]

Common Morphological Operations

• Erosion:
  • Removes pixels on object boundaries.
  • Shrinks the size of objects in the image.
  • Used to eliminate small noise or detach connected objects.
• Dilation:
  • Adds pixels to object boundaries.
  • Enlarges the object in an image.
  • Helps fill small holes and gaps within objects.
• Opening:
  • Erosion followed by dilation.
  • Used to remove small objects (noise) while maintaining the shape of larger objects.
• Closing:
  • Dilation followed by erosion.
  • Fills small holes and gaps in an object’s boundaries.
• Top-Hat Transformation:
  • The difference between the original image and its opening.
  • Used to highlight bright regions on a dark background.
  • Detecting small, bright objects or details in an unevenly illuminated image.
• Black-Hat Transformation:
  • The difference between the closing of an image and the original image.
  • Used to highlight dark regions on a bright background.
  • Emphasizing dark objects or shadows in an image.

Example of Morphological Operations

Introduction to Frequency Response

• What is Frequency Response?
  • The frequency response of an image shows how spatial details in the image are distributed across different frequencies.
  • In image processing, this is typically analyzed using the Fourier Transform.
• Why Frequency Analysis?
  • Useful for identifying patterns, noise, and image structures not easily observed in the spatial domain.

The Fourier Transform

• Fourier Transform (FT):
  • Converts an image from the spatial domain (pixels) to the frequency domain (sinusoids).
  • Each point in the frequency domain represents a specific frequency in the image.
• Mathematical Basis:
  • \(F(u,v) = \sum_x \sum_y f(x,y) e^{-j 2 \pi (ux/M + vy/N)}\)
  • Where \(F(u,v)\) is the frequency representation of the image.

Low and High Frequencies

• Low Frequencies:
  • Represent slow variations or large structures in the image (e.g., background or smooth gradients).
• High Frequencies:
  • Represent rapid variations or fine details (e.g., edges, noise).
• Key Insight: Most of the important structural information in an image is captured in the low-frequency range.

Frequency Domain Representation

• The Fourier Transform of an image produces a frequency spectrum.

• DC Component (center of the spectrum): Represents the average intensity of the image.

• Higher frequencies: Spread out from the center and capture finer details.

• Logarithmic scale: Often used to visualize the frequency spectrum due to the wide range of values.

The 2D Discrete Fourier Transform (DFT)

• The 2D DFT is used to convert a 2D image into its frequency components:
  • Input: A 2D grayscale image.
  • Output: A complex matrix representing amplitude and phase for each frequency.
• Inverse DFT: Converts the frequency representation back to the spatial domain.

Low-Pass and High-Pass Filtering

• Low-Pass Filter (LPF):
  • Allows low frequencies to pass, attenuates high frequencies.
  • Used to blur images, removing high-frequency details like noise and edges.
• High-Pass Filter (HPF):
  • Allows high frequencies to pass, attenuates low frequencies.
  • Used to sharpen images by enhancing edges and fine details.

Band-Pass Filtering

• Band-Pass Filter:
  • Allows frequencies within a certain range (band) to pass.
  • Useful for selectively enhancing specific frequency components while filtering others.
• Applications: Used in image enhancement and texture analysis.

Frequency Response Visualization

• Magnitude Spectrum:
  • Represents the amplitude of each frequency component.
  • Typically visualized using the logarithmic scale to manage the large range of values.
• Phase Spectrum:
  • Represents the phase of each frequency component.
  • Less important for human perception but crucial for reconstructing the image.

Applications of Frequency Domain Processing

• Noise Removal: Low-pass filters can smooth out high-frequency noise.
• Edge Detection: High-pass filters enhance edges and sharp transitions.
• Image Compression: Frequency domain analysis helps identify redundant information.
• Pattern Recognition: Useful for detecting repetitive patterns like textures.